Concession and contrast
Concession (la concession) admits an obstacle and says in the same breath that it made no difference: Although it's raining, I'm going out. Contrast (l'opposition) simply sets two facts side by side, with neither getting in the way of the other: I like the south, whereas my sister prefers the north.
Concession and contrast by ear
Bien qu'il pleuve, je sors.
Although it's raining, I'm going out.
Il est venu malgré la pluie.
He came despite the rain.
J'aime le Sud, alors que ma sœur préfère le Nord.
I like the south, whereas my sister prefers the north.
bien que or même si?
Both conjunctions concede something, but they say different things about the obstacle. quoique is the more literary twin of bien que. Both elide before a vowel: bien qu'il, quoiqu'elle.
bien que against même si
bien que / quoique + subjunctive
The obstacle is a fact that is being granted. The subjunctive is demanded by the conjunction; it does not signal doubt. English uses the indicative here, so nothing in your ear warns you.
Bien que je sois fatigué, je finirai ce dossier.
Although I'm tired, I'll finish this file.
Elle a accepté, bien qu'elle ait peu de temps.
She said yes, even though she has little time.
Quoiqu'il fasse froid, la terrasse est pleine.
Although it's cold, the terrace is full.
même si + indicative
Whether the obstacle is real is left open (even if), with the indicative. With the imparfait in the si-clause and the conditionnel in the main clause, the case becomes unreal.
Même s'il est malade, il travaillera.
Even if he is ill, he will work.
Même si tu insistais, je refuserais.
Even if you insisted, I would say no.
en dépit de
en dépit de means the same as malgré (in spite of), takes a noun too, and sounds more written. English follows despite with an -ing form (despite being tired); French cannot do that. malgré que is treated as incorrect: for a clause, use bien que.
En dépit de ses efforts, le projet a échoué.
In spite of his efforts, the project failed.
Elle a signé en dépit des avertissements de son avocat.
She signed in spite of her lawyer's warnings.
avoir beau + infinitive
avoir beau + infinitive means however much I do X, it makes no difference; English needs no matter how hard, however often or try as I might. Only avoir is conjugated, and beau never changes: elle a beau essayer, ils ont beau essayer. The main clause follows straight after, without mais.
The pattern
J'ai beau lui expliquer, il ne comprend pas.
However often I explain it to him, he doesn't understand.
Elle a beau être timide, elle parle très bien en public.
Shy as she is, she speaks very well in public.
Nous avons eu beau chercher, nous n'avons rien trouvé.
However long we searched, we found nothing.
alors que and tandis que
alors que and tandis que put two facts against each other and take the indicative. Both can also be purely temporal (while); context decides.
Mon frère adore la ville, alors que moi, je préfère la campagne.
My brother loves the city, whereas I prefer the countryside.
Elle aime le thé, tandis que lui ne boit que du café.
She likes tea, whereas he only drinks coffee.